Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T19:32:14.616Z Has data issue: false hasContentIssue false

Recollement of homotopy categories and Cohen-Macaulay modules

Published online by Cambridge University Press:  04 November 2011

Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya, [email protected]
Kiriko Kato
Affiliation:
Graduate School of Science, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, [email protected]
Jun-ichi Miyachi
Affiliation:
Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, [email protected]
Get access

Abstract

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.

Type
Research Article
Copyright
Copyright © ISOPP 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A1.Auslander, M., Representation dimension of artin algebras, Queen Mary College Mathematics Notes, 1971.Google Scholar
A2.Auslander, M., Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), pp. 1244, Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978.Google Scholar
ABr.Auslander, M., Bridger, M., Stable module theory, Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence, R.I. 1969.Google Scholar
ABu.Auslander, M., Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay, 1987). Mem. Soc. Math. France (N.S.) 38 (1989), 537.Google Scholar
AR1.Auslander, M., Reiten, I., Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111152.Google Scholar
AR2.Auslander, M., Reiten, I., Cohen-Macaulay and Gorenstein Artin algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 221245, Progr. Math. 95, Birkhauser, Basel, 1991.Google Scholar
BBD.Beilinson, A. A., Bernstein, J. and Deligne, P., Faisceaux Pervers, Astérisque 100 (1982).Google Scholar
Bu.Buchweitz, R.O., Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, Unpublished manuscript (1987), 155 pp.Google Scholar
Ch.Chen, X., The Stable Monomorphism Category of a Frobenius category, arXiv:0911.1987.Google Scholar
Cr.Christensen, L. W., Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer-Verlag, Berlin, 2000.Google Scholar
CR.Curtis, C.W., Reiner, I., Methods of representation theory. Vol. I. With applications to finite groups and orders. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1990.Google Scholar
EJ.Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics 30 (2001).Google Scholar
H1.Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988.Google Scholar
H2.Happel, D., On Gorenstein algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 389404, Progr. Math. 95, Birkhauser, Basel, 1991.Google Scholar
Iw.Iwanaga, Y., On rings with finite self-injective dimension, Comm. Algebra 7 (1979), no. 4, 393414.CrossRefGoogle Scholar
I.Iyama, O., Auslander-Reiten theory revisited. Trends in representation theory of algebras and related topics, 349397, EMS Ser. Congr. Rep., Eur.Math. Soc., Zürich, 2008.Google Scholar
IKM.Iyama, O., Kato, K., Miyachi, J., Polygon of recollements, in preparation.Google Scholar
IK.Iyengar, S., Krause, H., Acyclicity versus total acyclicity for complexes over noetherian rings, Documenta Math. 11 (2006), 207240.CrossRefGoogle Scholar
JK.Jørgensen, P., Kato, K., Symmetric Auslander and Bass categories, arXiv:1001.3066.Google Scholar
Ke.Keller, B., Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379417.Google Scholar
Kr.Krause, H., The stable derived category of a noetherian scheme, Compos. Math. 141 (2005), 11281162.CrossRefGoogle Scholar
LZ.Li, Z., Zhang, P., Gorenstein algebras of finite Cohen-Macaulay type, preprint.Google Scholar
Mi1.Miyachi, J., Localization of Triangulated Categories and Derived Categories, J. Algebra 141 (1991), 463483.CrossRefGoogle Scholar
Mi2.Miyachi, J., Injective Resolutions of Noetherian Rings and Cogenerators, Proceedings of The AMS 128 (2000), no. 8, 22332242.CrossRefGoogle Scholar
Ne1.Neeman, A.The derived category of an exact category, J. Algebra 135 (1990), 388394.CrossRefGoogle Scholar
Ne2.Neeman, A. Triangulated Categories, Annals of Math. Studies 148, Princeton Univ. Press, 2001.Google Scholar
O1.Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, arXiv: math/0503632.Google Scholar
O2.Orlov, D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227248.Google Scholar
Rd.Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303317.CrossRefGoogle Scholar
Y.Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series 146, Cambridge University Press, Cambridge, 1990.Google Scholar